On the Bohr-Sommerfeld Quantization in the Pseudoclassical Systems

Abstract

The pseudoclassical mechanics') is known as a convenient method to deal with spinning particles classically. The main feature of this theory is introducing anti-commuting (Grassmann) vari­ ables which are classical counterparts of q-num­ ber fermionic variables. In the framework of this theory, Dirac particles can be treated as a dynamical system having the supersymmetry between four-vector Grassmann variables and position variables of the particle. When we apply the pseudoclassical mechanics to the problem of bound states of Dirac particles as in the quark model of hadrons, it becomes sometimes necessary to solve the energy eigenvalue semiclassically. The purpose of this paper is to investigate the semiclassical quantiza­ tion of Grassmann variables 2 ) which corresponds to the Bohr-Sommerfeld quantization of the usual canonical variables. Let us consider the dynamical system consisting of Grassmann variables ~i (i = 1, 2, ... , N) and the usual dynamical variables qa'S. We assume that the Lagrangian is given by3)